Spin-statistics theorem

The spin-statistics theorem associates integer-spin particles with bosonic exchange symmetry and half-integer-spin particles with fermionic exchange symmetry in relativistic quantum field theory under the standard locality and positivity assumptions.

A spin-one parent cannot decay into two identical spin-zero particles in the same internal state while preserving rotational symmetry and Bose exchange symmetry. Their final total angular momentum is orbital $L=1$, which is antisymmetric under interchange. Bose symmetry requires even $L$ instead. An energetically allowed decay with matching parity can therefore remain forbidden independently of the interaction's coupling strength.

Restrictions linking the exchange symmetry of nuclear spin and spatial wavefunctions. For identical nuclei, the total wavefunction must be antisymmetric for fermions and symmetric for bosons. Electronic and vibrational exchange parity must be included before assigning rotational parity.

Exchanging two identical bound atoms exchanges their constituent fermions. An atom with mass number $A$ and atomic number $Z$ contains $A$ nucleons and $Z$ electrons, so its exchange sign is $(-1{)}^{A+Z}$ when its internal state is preserved. An odd count gives a composite fermion; an even count gives a composite boson. This is equivalently the integer-versus-half-integer total-spin distinction in the Spin-statistics theorem. For example, neutral lithium-6 has nine fermionic constituents and is a fermion, while neutral lithium-7 has ten and is a boson. Counting nucleons alone instead classifies the nucleus, not the neutral atom. This effective composite-particle description requires energies and separations where the bound atom remains intact.